Stability of switched systems with multiple equilibria: a mixed stable-unstable subsystem case

This paper studies the stability of switched systems that are composed of a mixture of stable and unstable modes with multiple equilibria. The main results of this paper include some sufficient conditions concerning set convergence of switched nonlinear systems. We show that under suitable dwell-time and leave-time switching laws, trajectories converge to an initial set and then stay in a convergent set. Based on these conditions, Linear Matrix Inequality (LMI) conditions are derived that allow for numerical validation of the practical stability of switched affine systems, which include those with all unstable modes. Two examples are provided to verify the theoretical results

Stabilization of systems with switchings on the axis of their coordinates and its input-to-state properties

International audienceThe stabilization problem for switched systems in which switchings occur on the axes of its state coordinates is considered. It is shown that a linear feedback, or a combination of linear feedback and a switching law, can be designed such that the closed-loop is stable, and has the input-to-state property, allowing to guarantee robustness against matched and unmatched perturbations. The conditions of stability are expressed in the form of linear matrix inequalities. The results are illustrated by numerical simulations

Relay Control Design using Attractive Ellipsoids Method

International audienceBased on the attractive ellipsoids method the problem of rejection of both matched and mismatched exogenous disturbances is studied for a linear plants controlled by the generalized relays. Some existing results about local stabiliz-ability as well as matching condition are refined for the system under consideration. The procedure of robust control design for optimal rejection of bounded exogenous disturbances is proposed. The theoretical results are supported with numerical simulations

Stabilization by a relay control using non-quadratic Lyapunov functions

International audienceIn this article we consider the stabilization problem by a relay control using non-quadratic Lyapunov functions. First, a general result is proposed for the case of nonlinear systems. A full state relay feedback controller is designed in order to ensure the local asymptotic stability of the closed-loop system. Then, the result is applied to the particular case of LTI systems. A constructive method based on LMI conditions is given in order to design nonlinear switching surfaces and provide an estimation of a non-ellipsoidal domain of attraction. In addition, the approach is extended to observer-based relay feedback. Both linear and nonlinear switching surfaces dependent on the estimated state are designed while using a Luenberger observer. Finally, illustrative examples are proposed in order to show the efficiency of the proposed methods and simulations are performed for a Buck converter structure

Contributions to Control of Electronic Power Converters

This thesis deals with the control of electronic power converters. In its development two main parts have been differentiated. On the one hand, the problem of the voltage balance in the capacitors of the dc-link in a three-level NPC converter is addressed. On the other hand, given that the techniques used in the first part to model the converters need to make certain assumptions and, with the intention of avoiding averaged models, in the second part, switched affine models have been developed to design the control of the output voltage in DC-DC boost type converters. In this way, in the first part several control laws have been developed using an averaged model formulated by duty cycles for each level in each phase. This formulation allows to consider, in the controllers design stage, the degree of freedom associated with the homopolar voltage injection. Therefore, the controllers are designed as well as a part of the modulation, so that control and modulation are integrated in the same stage. In this way, three controllers have been designed where, apart from the objective of the voltage balance of the capacitors, other objectives such as the number of commutations or the quality of the output signal have also been improved. In the second part of the thesis, four methods have been developed for the design of control laws taking advantage of the modeling of converters as switched affine systems given their hybrid behaviour. Thus, the first two laws take advantage of this modeling using the delta operator to avoid numerical problems when using systems where the sampling time is very low. The first of these controllers is based on Lyapunov’s function while the second is independent of this function, thus obtaining less conservative results. The other two laws developed for switched affine systems use an alternative model to that performed in the first two controllers, so certain existing disadvantages are avoided using again a design not based on Lyapunov’s function. Thus, the first law presents a basic control but, even so, improves the results of other existing laws in the literature. Finally, a design method to deal with systems with variations in their parameters has been presented.La presente tesis trata sobre el control de convertidores electrónicos de potencia. En su desarrollo se han diferenciado dos partes principales. Por un lado, se trata el problema del balance de tensiones en los condensadores que forman el dc-link en un convertidor NPC de tres niveles. Por otro lado, dado que las técnicas utilizadas en la primera parte para modelar los convertidores necesitan realizar determinadas suposiciones y, con la intención de evitar modelos promediados, en la segunda parte se han desarrollado modelos afines conmutados para diseñar el control de la tensión de salida en convertidores DC-DC tipo boost. De esta forma, en la primera parte se han desarrollado varias leyes de control utilizando un modelo promediado formulado mediante ciclos de trabajo para cada nivel en cada fase. Esta formulación permite considerar en la fase de diseño de los controladores, un grado de libertad asociado a la inyección de tensión homopolar. Por lo tanto, se diseñan los controladores a la vez que una parte de la modulación, de forma que se integra control y modulación en una misma fase. De esta forma, se han diseñado tres controladores donde, a parte del objetivo de balancear la tensión de los condensadores, se ha ido buscando mejorar también otros objetivos como el número de conmutaciones o la calidad de la señal de salida. En la segunda parte de la tesis, se han desarrollado cuatro leyes de control aprovechando el modelado de convertidores como sistemas afines conmutados dada su naturaleza híbrida. De esta forma, las dos primeras leyes, aprovechan dicho modelado usando el operador delta para evitar problemas numéricos al utilizar sistemas donde el tiempo de muestreo es muy bajo. El primero de dichos controladores está basado en la función de Lyapunov mientras que el segundo es independiente de dicha función obteniendo así resultados menos conservadores. Las otras dos leyes desarrolladas para sistemas afines conmutados utilizan un modelado alternativo al realizado en las dos primeras, de forma que se evitan ciertas desventajas existentes y mantienen un diseño no basado en la función de Lyapunov. Así, la primera ley presenta un control más básico pero que, aun así, mejora los resultados de otras leyes existentes en la literatura. Por último, se ha presentado un procedimiento de diseño que hace frente a sistemas con variaciones en sus parámetros

Contribuições à teoria de controle de sistemas afins com comutação com aplicações em eletrônica de potência

Orientador: Grace Silva DeaectoTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia MecânicaResumo: Esta tese é dedicada ao estudo da teoria de controle de sistemas afins com comutação e algumas de suas aplicações no contexto de eletrônica de potência. Após discussões preliminares, as contribuições principais são apresentadas. O objetivo comum ao longo deste trabalho é desenvolver, sob a perspectiva de otimização convexa, estratégias capazes de governar eventos de chaveamento em sistemas dinâmicos afins de maneira a levar a trajetória do estado a um ponto de referência desejado ou a rastrear uma trajetória variante no tempo. Metodologias de projeto, baseadas em uma função de Lyapunov quadrática generalizada, para função de comutação dependente do estado ou da saída são fornecidas para sistemas afins com comutação a tempo discreto para os quais apenas estabilidade prática é possível de ser assegurada. Subsequentemente, novas condições para estabilidade prática são introduzidas baseadas em desigualdades de Lyapunov-Metzler e levando em conta uma função de Lyapunov do tipo mínimo, que permite reduzir o conservadorismo referente à garantia de estabilidade. Uma metodologia para projetar ciclos limites e assegurar a estabilidade assintótica global foi também apresentada, que leva em conta uma função de Lyapunov variante no tempo e permite tratar otimização de desempenho H2 e Hinf. Ademais, novas discussões sobre a estabilidade de uma classe de sistemas com comutação não-lineares a tempo contínuo são introduzidas, nas quais o problema de rastreamento de trajetória é tratado. O estudo desta classe é de interesse visto que ela modela o comportamento dinâmico de conversores de potência CA-CC e de máquinas síncronas de ímã permanente alimentadas por inversores de tensão. Esta nova abordagem permite o controle de forma mais simples quando comparada a estratégias clássicas de controle vetorial. Finalmente, alguns resultados experimentais são apresentados, validando as estratégias de controle desenvolvidas. As condições de estabilidade e projeto são majoritariamente escritas em termos de desigualdades matriciais lineares e, logo, podem ser resolvidas de forma eficiente por resolvedores de programação semi-definida prontamente disponíveisAbstract: This dissertation is devoted to the study of switched affine systems control theory and some of its applications in power electronics context. After some preliminary discussions, the main contributions are presented. The common goal throughout this work is to develop, from a convex optimization viewpoint, strategies capable of governing switching events in dynamical affine systems in order to bring the state variable to a desired reference value or to track a time-varying trajectory profile. Design methodologies for state or output dependent switching function based on a generalized Lyapunov function are provided for discrete-time switched affine systems, where only practical stability is possible to be assured. Subsequently, novel practical stability conditions are proposed, based on Lyapunov-Metzler inequalities and taking into account a min-type Lyapunov function, which allows us to reduce conservativeness regarding stability guarantee. A methodology for designing limit cycles and assuring their global asymptotic stability is also presented, which takes into account a time-varying Lyapunov function and permits to cope with H2 and Hinf performance optimization. Afterward, novel discussions on the stability of a continuous-time nonlinear switched systems class are introduced, where the trajectory-tracking problem is addressed. The study about this class is of interest as it models the dynamic behavior of AC-DC power converters and permanent magnet synchronous machines fed by voltage source inverters. This new approach allows their control in a simpler manner when compared to classical field-oriented control strategies. Finally, some experimental results are presented, validating the developed control strategies. Stability and design conditions are mostly written as linear matrix inequalities and, thus, can be efficiently solved by readily available semi-definite programming solversDoutoradoMecatrônicaDoutor em Engenharia MecânicaPDSE 88881.187487/2018-01CAPESCNPQFAPES

Local stabilization of switched affine systems

International audienceThis paper considers the local stabilization problem for the class of switched affine systems. The main idea is to use an alternative representation of the switched affine system as a nonlinear system with input constraints. Switching laws can be derived by emulating locally classical controllers. It is shown that by restricting to local stabilization, the classical constraint on the existence of constant stable convex combinations may be easily avoided. The approach may be interpreted as a generalization where convex combinations defined as functions of the system state are being used. Constructive methods for deriving switching laws are proposed